3.1348 \(\int \frac{1}{(c+d x)^2} \, dx\)

Optimal. Leaf size=12 \[ -\frac{1}{d (c+d x)} \]

[Out]

-(1/(d*(c + d*x)))

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Rubi [A]  time = 0.0016551, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ -\frac{1}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(-2),x]

[Out]

-(1/(d*(c + d*x)))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x)^2} \, dx &=-\frac{1}{d (c+d x)}\\ \end{align*}

Mathematica [A]  time = 0.0023961, size = 12, normalized size = 1. \[ -\frac{1}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(-2),x]

[Out]

-(1/(d*(c + d*x)))

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Maple [A]  time = 0.001, size = 13, normalized size = 1.1 \begin{align*} -{\frac{1}{d \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2,x)

[Out]

-1/d/(d*x+c)

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Maxima [A]  time = 0.961277, size = 16, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2,x, algorithm="maxima")

[Out]

-1/((d*x + c)*d)

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Fricas [A]  time = 1.56419, size = 24, normalized size = 2. \begin{align*} -\frac{1}{d^{2} x + c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2,x, algorithm="fricas")

[Out]

-1/(d^2*x + c*d)

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Sympy [A]  time = 0.291893, size = 10, normalized size = 0.83 \begin{align*} - \frac{1}{c d + d^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2,x)

[Out]

-1/(c*d + d**2*x)

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Giac [A]  time = 1.08254, size = 16, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (d x + c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2,x, algorithm="giac")

[Out]

-1/((d*x + c)*d)